7_General Equilibrium Under Uncertainty

8/19/2019 7_General Equilibrium Under Uncertainty

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7. General Equilibrium underUncertainty

Uncertainty is formalized by the existence of some states of the world s =1, ...,S. Technologies, endowments and preferences depend on the state of theworld. Assume that S is finite and that different states are mutually exclusive.

A state-contingent commodity vector is x = (x11,...,xL1,...,x1S ,...,xLS ) ∈RLS is an entitlement to receive commodity vector (x1s,...,xLs) if state s ∈ S

occurs. Endowments are also contingent: ωi = (ω11,...,ωL1,...,ω1S ,...,ωLS ) ∈RLS .

Preferences are also contingent on the state of the world: i on X i ⊂ RLS .

The most usual example is that of Bernoulli utility functions: usi(x1s,...,xLs)with x

i

i x

i if, and only if,

s

πsi

usi

(x1s

,...,xLs

) ≥s

πsi

usi

(x

1s,...,x

Ls).

For each state, agents have utility functions. An agent is risk-averse if hisutility function is concave, and risk neutral if it is linear. Concave Bernoulliutility functions imply convex preferences. With one good and two states of theworld, the slope at any point along an indifference curve is given by:

M RS = π1iu

i(x1)

(1 − π1i)ui(x2)

The set of points on the diagonal, i.e. such that x1 = x2 is the certaintyline. Along those points, the individual faces no risk.

With respect to the production side of the economy, we consider state-contingent production plans yj ∈ RLS . Shares θij ≥ 0 are not state-contingent.

1 Arrow-Debreu equilibrium

If at date zero, before the resolution of uncertainty, there is a market for everycontingent commodity ls, then we can introduce the concept of Arrow-Debreuequilibrium as the relevant equilibrium concept. At date zero, what is beingtraded are commitments to receive or deliver amounts of physical good l if states occurs. Deliveries are contingent on the state, but payments are not. Afteruncertainty is solved, and the agents know the realization of the state, what theypromised to deliver and/or receive at any other state of the world is irrelevant.

The definition of an Arrow-Debreu equilibrium is an allocation (x∗1,...,x∗I , y∗1 ,...,y∗J ) ∈RLS (I +J ) and p = ( p11,...,pLS ) ∈ RLS such that:

i) ∀ j, y∗j satisfies py

∗j ≥ pyj ∀yj ∈ Y j .

ii) ∀i, x∗i is maximal for i in

xi ∈ X i such that pxi ≤ pωi +

j

θij py∗j

.

iii)i

x∗i =j

y∗j +i

ωi.

If such date zero (before the resolution of uncertainty) markets exist, the twowelfare theorems apply. Thus, an efficient allocation of risk can be achieved.

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In order to satisfy convexity of preferences, we need to assume that the utility

functions be concave.For example, in an economy with two agents, one physical good, and twostates of the world and endowments ω1 = (1, 0) and ω2 = (0, 1), if utilityfunctions are U i(x1i, x2i) = π1iui(x1i) + (1 − π1i)ui(x2i), for i = 1, 2, Arrow-Debreu equilibrium will imply that the final allocation is an efficient distributionof risk, in other words, that the final allocation will be in the Pareto Set. This is just a direct application of the First Welfare Theorem. Moreover, if π11

π21= π12

π22,

then the Pareto set will be the diagonal.In the case of aggregate risk existing, i.e., if total endowment is different

across states of the world, then the price of a unit of the good delivered in thestate where total endowment is lower is higher than in the other state (assumingboth states happen with the same probability and both agents have the sameexpectations on the probabilities).

Example 1 Consider an economy with many identical consumers. The econ-omy lasts two periods. The consumers have identical endowments of trees, ω.

Each tree yields f 1 units of fruit in period 1 and f 2 units of fruit in period 2.Each consumer has utility function u(x1, x2), where xi is fruit eaten in period i.There are competitive markets in trees, fruit in period 1 and fruit in period 2,and all clear before period 1. Derive the equation for the price of trees relative to the price of fruit in terms of the preferences and endowment.

Let t be the number of trees purchased, and z1, z2 the amounts of fruit pur-chased in excess of what the trees purchased produce. Then, the consumer’s problem is

maxz1,z2,t≥0

u(f 1t + z1, f 2t + z2)

s.t. p1z1 + p2z2 + ptt ≤ ptω,

and therefore, the first-order conditions, at an interior solution, are:

u1 = λp1

u2 = λp2

u1f 1 + u2f 2 = λpt

Thus, we obtain:

u1

u2=

p1

p2 pt = p1f 1 + p2f 2,

which implies that, if we normalize by dividing by the price of fruit in the second period,

pt

p2=

u1

u2f 1 + f 2.

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Example 2 Consider a competitive economy with two periods and two states

of nature. The probability of state 1 is 1/3, and that of state 2 is 2/3. There are three goods in the economy: a is first period consumption, b1 is second period state 1 consumption, and b2 is second period state 2 consumption. There is a continuum of measure 1 of identical consumers, who are expected utility maximizers, using correct probabilities, and whose utility function is u = 1

2 ln a+12

ln b. The endowments are the same for every consumer: 12 units of the first period consumption good and none of the second period consumption good.

There are two types of firms, and a large number of each type (the exact number is irrelevant, since there are CRS). All firms have CRS. Type 1 firms can convert 1 unit of first period consumption into 1.25 units of second period state 1 consumption. Type 2 firms can convert 1 unit of first period consumption into0.5 units of second period state 1 consumption, and 0.5 units of second period state 2 consumption. Each firm is owned in equal shares by all consumers (since

profits will be zero, the ownership structure is irrelevant).If there is a complete set of contingent markets, what are the Walrasian

Equilibrium price and quantities?

2 Radner equilibrium

The next step is to consider what happens if we remove the assumption of forward markets existing at time zero. Assume that t = 0, 1. There is no infor-mation at t = 0, and at t = 1, uncertainty is resolved. From the First WelfareTheorem, it is clear that agents have no incentive to retrade on spot marketsat t = 1, after resolution of uncertainty. However, if not all the LS contingentmarkets exist, it could still be the case that Pareto optimality can be reached.

In order for this to happen, we need consumers to be able to somehow transferwealth across states. This is the case when at least one physical commoditycan be traded contingently at t = 0, spot markets open at t = 1, and agentscorrectly anticipate spot prices at t = 0.

This way, consumers solve

maxxi∈RLS

U i(x1i,...,xSi)

s.t.s

q szsi ≤ 0 and psxsi ≤ psωsi + p1szsi∀s

Now, zsi is good one traded contingently at time zero, and the price of thisgood is q s. Notice that there are S + 1 constraints in total, one for contingent

trade at t = 0, and one for each of the states of the world at t = 1. Of course,once uncertainty is resolved, only one of the S constraints that are included inthe second condition will actually be in place, the rest of the constraints will beirrelevant, because those states of the world will never actually happen.

The equilibrium concept in this setting is Radner equilibrium. We will definewhat a Radner equilibrium is and then argue that it is equivalent to the Arrow-Debreu equilibrium.

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q ∈ RS , ps = ( p1s,...,pLs) ∈ RL ∀s, z∗i ∈ RS ∀i, and x∗

i ∈ RLS ∀i constitutes

a Radner equilibrium if i) ∀i, z∗i , x∗i solve the above utility maximization problem.ii)i

z∗si ≤ 0,i

x∗si ≤i

ωsi ∀s.

It turns out that the set of Arrow Debreu and Radner equilibrium allocationsare identical. Formally, this is presented in Proposition 19.D.1 in MWG, whichsays that if x∗ ∈ RLSI , p ∈ RLS

++ are an Arrow-Debreu equilibrium, ∃q ∈ RS ++,

z∗ ∈ RSI such that x∗, z∗ and ( p1,...,pS ) ∈ RLS ++ are a Radner equilibrium.

Conversely, if x∗ ∈ RLSI , z∗ ∈ RSI , q ∈ RS ++ and ( p1,...,pS ) ∈ RLS

++ are aRadner equilibrium, then ∃ (µ1,...,µS ) ∈ RS

++ such that x∗, (µ1 p1,...,µS pS ) ∈RLS ++ are an Arrow-Debreu equilibrium. The multiplier µs is the value, at t = 0,

of a dollar at t = 1 and state s, or the relative value of wealth at t = 1.Those multipliers will be chosen so that µs p1s = q s. It is important that agents

correctly anticipate spot prices at t = 0.

3 Assets

Assets play the role of transferring wealth across states of the world (or acrosstime), in a similar way as contingent trade in good 1. A unit of an asset is anentitlement to receive an amount rs of good 1 at t = 1 if state s occurs. Thus,we can characterize an asset by its return vector r = (r1,...,rS ) ∈ RS . Examplesof assets are Arrow security: (0, 0, 0, 1, 0) or a riskless asset: (1, 1, 1, 1).

We assume there is an exogenously given set of K assets, which will consti-tute the asset structure. These assets have prices (q 1,...,q K ). We thus generalizethe definition of a Radner equilibrium given the asset structure.

q ∈ RK

, asset prices at t = 0, ps = ( p1s,...,pLs) ∈ RL

∀s, z∗

i = (z∗

1i,...,z∗

Ki)at t = 0 and x∗i ∈ RLS at t = 1 are a Radner equilibrium if

a) ∀i, z∗i , x∗i solve:max

xi ∈ RLS

zi ∈ RK

U i(x1i,...,xSi)

s.t. i)k

q kzki ≤ 0

ii) psxsi ≤ psωsi +k

p1szkirsk ∀s

b)i

z∗ki ≤ 0 ∀k,i

x∗si ≤i

ωsi ∀s.

Thus, given the asset structure of the economy, we can define the returnmatrix R whose kth column is the return vector of the kth asset. If rank(R) = S ,

then the asset structure is said to be complete. If rank(R) < S, the assetstructure is incomplete.

Using this notation, the budget constraint becomes:

Bi( p, q, R) =

x ∈ RLS s.t. for some portfolio zi ∈ R

K , q zi ≤ 0 and

p1(x1i − ω1i)

...

pS (xSi − ωSi)

≤ Rzi

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If the asset structure is complete, then

i) if x

∗

∈ RLSI

and p ∈ RLS

++ are an Arrow-Debreu equilibrium, then ∃ assetprices q ∈ RK ++ and z∗ = (z∗1 ,...,z∗I ) ∈ R

KI such that x∗, z∗ q , ( p1,...,pS ) ∈ RLS ++

are a Radner equilibrium.ii) if x∗ ∈ RLSI , z∗ = (z∗1 ,...,z∗I ) ∈ RKI , q ∈ RK

++, ( p1,...,pS ) ∈ RLS ++ are a

Radner equilibrium, then ∃ (µ1,...,µS ) ∈ RS ++ such that x∗, (µ1 p1,...,µS pS ) ∈

RLS ++ are an Arrow-Debreu equilibrium.

Now, if the asset structure is not complete, or K < S , a Radner equilibriumneed not be Pareto optimal.

A way to price assets is by considering that q T = µ ·R (there are multipliersfor wealth that make this hold). In other words, the price must satisfy thearbitrage-free condition.

q ∈ RK ++ is arbitrage-free if z = (z1,...,zK ) such that q · z ≤ 0, Rz ≥ 0 and

Rz = 0.

Example 3 The arbitrage-free condition is a way to price an asset. For in-stance, in the following asset prices:

µ 100, 95ρ 95, 85

1 − µ 100, 9080, 70

λ 100, 901 − ρ 85, 75

1 − λ 100, 85

there are arbitrage opportunities. I can purchase one unit of asset 2 and sell 8595 units of asset 1. At t=2, if s=1, then I get 95, pay

8595100 ≈ 89.5. If s=2, I

get 90, while only paying 8595100 ≈ 89.5. There is no value of µ > 0 such that there are no arbitrage opportunities, i.e., such that

1, 85

95

=

1,

95

100µ +

90

100(1 − µ)

.

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7_General Equilibrium Under Uncertainty